Memarks on the Logic af Elementary Geometry. 368 



applicable to quantities whether commensurable or incommen- 

 surable. The former part ought therefore to be rejected. The 

 rule thus purified of extraneous matter will be — if, when equi- 

 multiples are taken of the alternate terms of four quantities, 

 it be found that those numbers which make the multiple of the 

 first greater than that of the second, do also make the multiple 

 of the third greater than that of the fourth, then these quan- 

 tities are proportionals ; or, if m A"7 n B when mC? n D '.' 

 A : B = C : D or li = D To proceed logically, we must of 

 course produce a demonstration of this, which is not difficult, 

 the converse however is much easier, and is all that seems to 

 be attempted in the elucidation of this subject, in the notes to 

 Playfair's Elements. 



It is well known that every theorem has its convsrse, in 

 which the conclusion of the theorem is converted into the hypo- 

 thesi"!, and the former hypothesis becomes the conclusion. By 

 this it is shown, that if a property belong to an object and dis- 

 tinguish and characterize it as an individual among other ob- 

 jects, that property cannot belong to these others. This then 

 necessarily foUows, that no theorem logically stated, can 

 be true without the converse also being true. The converse is 

 in this point of view, reduced to tlie clraracter of an identical 

 proposition with the theorem, and, except the peculiar importance 

 of the truth, and its frequent application, induce us to act 

 otherwise, ought always to be stated as a corollary to the 

 theorem. The proof will be indirect, and will consist of a 

 very short statement, showing that the theorem cannot be true 

 when the converse is denied. Apparent anomalies in regard to 

 the rule now stated, arise from illogical statements of the 

 theorems which seem to present them. In the hypothesis of a 

 theorem, no condition ought to be superfluous or unnecessary 

 as a foundation of the conclusion, and the conclusion ought to 

 be the whole truth which the demonstration makes to rest upon 

 the hypothesis. When any other particulars in either case are 

 rejected or admitted, the theorem is illogically stated, and 

 then the converse will, for that reason, be untrue. Of the 

 accuracy of this position we will easily be convinced by an 

 analysis of any theorem in which the converse seems to be un- 

 true — 1st. As to those in which only one geometrical fact is 

 used as the hypothesis, we have this example, "similar poly- 

 gons have their perimeters in the ratio of their homologous 

 sides," the converse of whic)i as thus stated is not true ; but 

 there is a superfluous condition in the hypothesis, viz : that which 

 relates to the angles. Excluding this condition, the theorem 

 may be stated thus : — When polygons have proportional sides, 

 their perimeters are as their homologous sides. Now, it is not 

 accurate to say — " Conversely : if perimeters of polygons have 



